Statistical Mechanics - Physics at Oregon State University

98 CHAPTER 5. FERMIONS AND BOSONS
Since this has to be identical to zero, all coefficients of all powers of t have to
be zero, and we have cn+2(n + 2)(n + 1) + cn = 0. We get two sets of solutions.
Either we set c0 = 1 and c1 = 0 or do the opposite. A general solution is a
linear combination of these two basic solutions. The basic solutions are
and
∞
k=0
∞
k=0
(−1) k
(2k)! t2k
(−1) k
(2k + 1)! t2k+1
(5.39)
(5.40)
which clearly converge for all values of time. Do these functions have special
properties? That is very hard to tell from the power series solutions. We
know however, that they are equal to cos(t) and sin(t) and are periodic. That
information comes from a different source of knowledge. We can also solve the
differential equation on a computer and plot the solution. Most likely we will see
the periodicity right away, but round-off errors will start to change the picture,
and perhaps we can see only a certain number of periods. Therefore, can we
conclude from the computer calculations that the solutions are periodic? Who
knows.
Another example is the differential equation
d3x + x = 0 (5.41)
dt3 It is easy to construct power series solutions, they are of the form:
∞
k=0
(−1) k
(3k)! t3k
and similar. Are they periodic? What other properties do they have?
Finding the chemical potential.
Once we know the grand energy we are able to calculate N and find
N(T, µ, V ) = −
∂Ω
∂µ T,V
= −
∂Ω
∂λ T,V
N = (2S + 1)V kBT λ −3
∂
T f 5
∂λ 2 (λ)
λ
kBT
∂λ
∂µ T,V
(5.42)
(5.43)
(5.44)
The derivative for our special function is easy to find for λ < 1 using the power
series.